A note on the growth of regularity with respect to Frobenius

Let $R=k[x_1,\dots ,x_n]\mathrm{/}I$ be a standard graded $k$-algebra where $k$ is a field of prime characteristic and let $J$ be a homogeneous ideal in $R$. Denote $(x_1,\dots ,x_n)$ by $\mathfrak{m}$. We prove that there is a constant $C$ (independent of $e$) such that the regularity of $H_{\mathfrak{m}}^s(R\mathrm{/}{J}^{[p^e]})$ is bounded above by $C{p}^e$ for all $e\ge 1$ and all integers $s$ such that $s+1$ is at least the dimension of the locus where $R\mathrm{/}J$ doesn’t have finite projective dimension.